yes, it is hygrophila angustifolia. it used grow alright but the stem slowly became dark and melted. the trimmed parts are still there, slow but growing.I'm going to guess your stunted stem plant in a hygro. angustifilia. It needs a fair amount of water column ferts & light. I've never grown it well.
Your sword...is that all you have left of it, those 2 , well, they're not rhizomes, but sort of, ahem, corms? I think it had no food it could use with maybe a bit of transplant shock. Are you using root tabs or other ferts? Swords are heavy pretty feeders...
Even java moss won't stay alive for too very long completely dried out...it's an aquatic plant! But I have had it recover from pretty dire conditions. It'll help if you can wet it or sprinkle some water on it often. Put in a jar/bowl?
The surface oil can be removed by gently laying a kitchen paper towel on it (or even newspaper) & removing it slowly. This will take several tries. How come you have oil? I don't remember, do you have fish & what are you feeding?
as for my sword, yes that is all that is left.